Investigation of Laser-Welded EH40 Steel Joint Stress with Different Thicknesses Based on a New Heat Source Model

Abstract

High-strength steel (HSS) plates are widely used due to their superior performance. However, residual stresses generated during welding can exacerbate the initiation of fatigue cracks, and the accurate prediction of residual stresses is crucial. Therefore, thermo-mechanical behavior analysis of the EH40 joints was completed based on the proposed new heat source model. The thermo-elastoplastic finite element analysis was determined via thermo-mechanical coupling with fully parametric programming. The influence of laser welding power and joint thickness on peak temperature and gradient was clarified. Meanwhile, it was found that when the laser welding power increased from 9 kW to 22.5 kW and the joint thickness increased from 6 mm to 15 mm, the distribution trend of longitudinal residual stress in the weld zone was gradually altered from a “U” shape to a “W” shape, while the transverse stress was transformed from a “U” shape to an “M” shape. It was determined that the amplitude of longitudinal and transverse stress changed along the thickness direction of nodes and was directly proportional to the peak temperature. The above results imply that the peak temperature, maximum temperature gradient longitudinal, and transverse residual stress distribution in the weld zone and its vicinity were remarkably affected by laser welding power and joint thickness.

1. Introduction

With the continuous development of aerospace, shipbuilding, nuclear industry, and steel-making technology, high-strength steel with high strength, good toughness, and weldability is widely used in these industries [1,2]. Due to the structural differences in various functional components in the equipment, HSS plates with different thicknesses are needed. Compared with laser multi-pass welding technology, single-pass full penetration laser welding with high efficiency and penetration is more attractive for welding HSS plates with different thicknesses [3]. However, the tensile residual stress (RS) [4,5] and inhomogeneous stress distribution produced in the process of welded joints in single-pass full penetration laser welding are known to seriously affect the mechanical properties of welded joints [6]. Therefore, it is necessary to study RS variation with the depth of different thickness steel joints in single-pass full penetration laser welding when inputting equal volume heat to provide useful guidance for their engineering application.

The heat source model reflects the distribution of heat input acting on the welded joint in both time and space domains, affecting the heat flow distribution during the welding process and significantly impacting the accuracy of the thermo-mechanical coupling numerical analysis of the welded joint. Therefore, in order to improve the prediction accuracy of temperature field in full penetration welding, scholars both domestically and internationally have attempted to use combined heat sources to describe their respective suitable welding scenarios. Liu et al. [7] proposed a finite element (FE) simulation process that combines two-dimensional and three-dimensional models to study the RS of 50 mm thick plates welded by electron beams. Their research results indicated that the simulation and experiment had good consistency, and the RS on the surface and weld profile had reasonable accuracy. Wang et al. [8] constructed a polynomial combined heat source model and found that the polynomial model achieved optimal consistency between simulated and experimental values, regardless of the cooling process, angular deformation, or predicted longitudinal and transverse residual stresses. Farroki et al. [9] developed a biconical volume heat source based on a three-dimensional conical (TDC) heat source, which was validated and calibrated through different experiments. Wu et al. [10] established a combined heat source model for the numerical simulation of plasma arc welding processes with small hole effects and large weld depth-to-width ratios. They found that under different welding conditions, the calculated weld geometry and fusion line trajectory at the cross-section maintained good consistency with experimental values. Li et al. [11] made an improved heat source model that considered the keyhole effect of plasma arc based on the actual shape of plasma arc welds. The improved heat source model and the consideration of fluid flow could improve the accuracy of PAW simulation, and it was found that the influence of fluid flow in the molten pool cannot be ignored. The effectiveness of the model was demonstrated by comparing simulation and experimental results. Kik et al. [12] established a new heat source model to modify the method of introducing welding material heat based on the difference in laser beam power distribution and shape. The model was calibrated and validated based on metallographic test results, actual thermal cycling during laser welding, and the obtained weld shape.

Many scholars have studied the RS of different thickness welded joints with single-pass and multi-pass welding through FE simulation calculation considering thermo-mechanical coupling and experimental. Liu et al. [13] studied the RS and microstructure of 6 kW laser welded joints of TC4 titanium alloy with 7.5 mm thickness and found that the columnar crystal structure at the upper molten pool was coarser than that of the lower, and there was a clear stress concentration at weld root and the weld toe. Fang et al. [14] conducted multi-layer and multi-pass laser metal deposition (4 kW) experiments on a 10 mm thick substrate (FV520B steel) with five material types and 3 mm thick stacking. The effects of preheating and phase transformation temperature on stress were deeply studied. It was observed that high or low phase transformation temperature greatly impacted tensile stress relaxation and inhomogeneous stress distribution, respectively. Chen et al. [15] used the thermo-elastoplastic finite element method (TEP-FEM) to simulate the RS of multi-layer and multi-pass welded (3.2 kW) butt joint of 10 mm thick 316L stainless steel. According to the stress distribution, it was observed that the compressive stress near the weld center and longitudinal stress reached self-equilibrating. Li et al. [16] reported the effect of precipitation strengthening behavior on welding residual stress during laser welding (9 kW) of 6 mm thick high-strength steel and found that it can slow down the high residual stress gradient values. Pu et al. [17] researched the deformation and RS distribution of 16 mm thickness HSS Q345 with multi-pass welding using three different modeling methods. The deformation and RS distribution of the sample were measured by three-coordinate measuring and hole-drilling methods. The simulation calculation and experimental showed that the coarse mesh density and instantaneous heat source model were more suitable for predicting the RS distribution of multi-pass welded joints. Jiang et al. [18] studied the effect of residual stress on hydrogen diffusion in 38 mm thick HSS plates using numerical simulation. A trustworthy numerical simulation approach, coupled with laboratory testing, was developed to assess the impact of residual stresses in HSS plates on hydrogen diffusion. Ghafouri et al. [19] studied the residual stress of 6 mm thick HSS T-welded joint S700 through numerical simulation and experimental verification using Goldak’s heat source model. It was discovered that the configuration of external constraints exerted a greater influence on sequential and cumulative welding distortions, as well as final residual stresses, compared to the welding sequences. Deep hole drilling (DHD) techniques were used to measure the RS in the through-thickness direction. They pointed out that the difference in strength between the base metal (BM) and the weld significantly affects the RS and distortion of the welded joint. Although the joints below 20 mm were mostly welded by multi-pass welding, the processing efficiency is much lower than that of single-pass welding. Lee et al. [20] and Gao et al. [21] used the TEP-FEM to study the RS of 6 mm dissimilar steel joints and aluminum alloy 2024 joints using single-pass welding, respectively. Rong et al. [22] employed hybrid laser–magnetic welding to conduct a single-pass laser (4 kW) welding experiment on 3.8 mm 316L steel and calculated the longitudinal and transverse RS of the welded joint. It was determined that the steady magnetic field was helpful in reducing distortion, transverse tensile stress, and longitudinal residual plastic strain in welded joints. Perić et al. [23] and Sun et al. [24], respectively, studied the RS distribution of 20 mm and 15 mm thick steel plate welded joints with single-pass welding through simulation and experimental verification. The former found that the transverse RS of the welded joint in the weld and its vicinity is close to 0 MPa, and the others are tensile stress. The latter’s simulation results showed that although the RS near the weld on the top surface of the welded joint is high, it still did not reach the yield strength of the material. Wu et al. [25] studied the RS distribution of 10 mm thick plate EH40 and 316L different butt joints in single-pass full penetration laser (10 kW) welding. The generation of transverse tensile stresses on the EH40 side was aggravated by the mismatch in material properties.

In conclusion, many studies have been carried out to accurately predict the RS of welded joints with various thicknesses through simulation or experiment. Previous research has been limited to the RS study of welded joints with a certain thickness. However, there is no research on the RS variation with the depth of different thickness welded joints from thin plate to thick plate in single-pass full penetration laser welding. Therefore, in this study, based on the principle that the equal volume heat and the same welding speed are input, HSS plates EH40 with different thicknesses (6 mm, 10 mm, and 15 mm) were welded using single-pass full penetration laser welding at different powers (9 kW, 15 kW, and 22.5 kW). A new heat source model is proposed to describe the weld morphology of high-power laser single-pass penetration welded joints of HSS plates with different thicknesses. The effects of laser welding power and thickness of welded joints on the peak temperature of the temperature field, as well as the stress field of the three welded joints with different thicknesses, were thoroughly studied based on the new heat source model, respectively. The whole calculation process was determined using ANSYS parametric design language (APDL); the purpose of this paper is to reveal RS variation with the depth of different thickness steel joints in single-pass full penetration laser welding when the equal volume heat and the same welding speed are input, to provide useful guidance for their engineering application.

2. Experiment

2.1. Welding and Measurement

The welding test material provided by WuYang Iron & Steel Co., Ltd. (Wugang, China) in this work was HSS plate EH40 (ASME SA-516 Gr.70) with different thicknesses (6 mm, 10 mm, and 15 mm), in which the length and width dimensions were 200 mm × 100 mm, respectively. The chemical composition of their base materials is shown in

Table 1. Chemical composition of steel plates EH40 with different thicknesses (wt.%).

	C	Si	Mn	P	S	Cu	Cr
6 mm	0.10	0.26	1.53	0.012	0.002	0.02	0.02
10 mm	0.141	0.16	1.45	0.019	0.005	0.01	0.017
15 mm	0.12	0.31	1.36	0.018	0.004	0.02	0.04
	Ni	Nb	V	Ti	Mo	Al	Fe
6 mm	0.01	0.027	0.005	0.013	0.003	0.029	Bal.
10 mm	0.007	0.041	0.004	0.017	0.002	0.027	Bal.
15 mm	0.02	0.01	0.003	0.012	0	0.021	Bal.

. In order to study the heat distribution and stress distribution of welded joints with different thicknesses under different welding power process parameters, as shown in Figure 1, high-power laser penetration welding was performed on the joints with different welding powers. In this paper, a 6-DOF KUKA robot was used to conduct a laser welding test with a 30,000-watt fiber laser system. The laser welding system mainly included a YLS-30000 fiber laser (Oxford, MI, USA), German KUKA six-axis robot (Augsburg, Germany), laser welding butt joint, shielding gas nozzle, clamp, and welding platform. According to previous research [1,2,3], it is known that various defects (such as incomplete penetration, porosity, spatter, etc.) are easily formed when using single-pass full penetration laser welding for thick plates. Therefore, after a large number of experiments, the welding parameters selected in this manuscript had good welding quality. The relevant welding experimental parameters were set separately. The welding power was 9 kW, 15 kW, and 22.5 kW, respectively; the welding speed was 1.8 m/min, the defocus was 0 mm, the shielding gas was argon, and the flow was 2.1 m³/h. In order to improve the stability and welding quality of the weld pool, acetone was used to clean the oil and oxide film on the surface of the joint before welding the high-strength steel joint.

As shown in Figure 2a, the PROTO iXRD equipment with the sin²Ψ method was used to measure RS in welded joints of HSS thick plates, where Ψ was the angle between the normal direction of the butt joints surface and the diffracted ray. During the measurement process, the Cr target was selected as the target material, and the X-ray source adopted the Cr-Kα. The diameter of the X-ray collimator’s spot was 1 mm, and the voltage and current were 20 kV and 4 mA, respectively. When measuring the surface RS of all welded joints in this paper, six points (P6~P11) on path L1, as shown in Figure 2b, were selected as measurement points. The coordinate positions of these six points along the Z-direction were 3 mm, 4 mm, 5 mm, −3 mm, −4 mm, and −5 mm, respectively. Simultaneously, the longitudinal and transverse measurements were repeated three times for each measuring point, and the average values were taken as the longitudinal and transverse RS values, respectively.

2.2. Weld Profile Analysis

In Figure 3a–c, the weld profiles of metallographic specimens with different thicknesses are shown. It can be observed that the entire weld consists of a BM, a heat-affected zone (HAZ), and a fusion zone (FZ). Moreover, with the increase in welding power and joint thickness, the weld morphology changes significantly. The weld gradually changes from the upper and lower sections of Figure 3a to the upper, middle, and lower sections of Figure 3b,c, and the waist of the middle section gradually forms a cylindrical shape. According to Figure 3, the existing simplified equivalent heat source models make it difficult to fully describe the weld morphology.

3. Thermo-Mechanical Model and Numerical Simulation

3.1. New Heat Source Model and Theoretical Derivation

Accurately predicting the temperature field of welded joints is crucial for accurately calculating subsequent thermo-mechanical coupled stress fields. According to the analysis of the metallographic test results in Figure 3, the existing heat source model makes it difficult to describe the actual weld morphology of high-power laser penetration welded joints of HSS thick plates. In view of this, a dual parabolic rotating body and cylindrical DPRC (Dual Parabolic Revolver Cylinder) composite heat source model, as shown in Figure 4, is proposed. The model mainly consists of three sections: upper, middle, and lower. The parabolic rotational model is used to describe the upper and lower segments, while the cylindrical model is used to describe the middle segment. The specific parameters of the composite model are the upper radius r_t and height z_i1 of the upper section of the parabolic rotating body, the radius r_i of the middle section of the cylindrical body, and the lower radius r_b and height z_i2 of the lower section of the parabolic rotating body.

In order to theoretically derive the composite heat source model, firstly, according to the actual distribution shape of the FZ in welding joints, it is assumed that the radius of the heat source decays in a parabolic pattern along the thickness direction of the workpiece (non-linear relationship). The radius of the upper, middle, and lower sections of the heat source can be as shown in Equation (1) to Equation (3):

$$r_1\left(z\right)={\displaystyle \frac{z^2}{p_1}}+k_1$$

(1)

$$r_2\left(z\right)={\displaystyle \frac{z^2}{p_2}}+k_2$$

(2)

$$r_3\left(z\right)={\displaystyle \frac{z^2}{p_3}}+k_3$$

(3)

where p_i and k_i represent intermediate variables.

According to the distribution characteristics of heat flux density mentioned above, the following energy conservation Equation (4) can be obtained:

$$\begin{array}{cc}\hfill Q=& {\displaystyle {\int }_{z_{i1}}^{z_t}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{r_1\left(z\right)}{q}_{m1}\left(r,z\right)}}}rdrd\theta dz+{\displaystyle {\int }_{z_{12}}^{z_{i1}}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{r_2\left(z\right)}{q}_{m2}\left(r,z\right)}}}rdrd\theta dz\hfill \\ & +{\displaystyle {\int }_{z_{i2}}^{z_b}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{r_3\left(z\right)}{q}_{m3}\left(r,z\right)}}}rdrd\theta dz\hfill \end{array}$$

(4)

where q_mi represents the energy distribution density.

The left side Q of the Equation (5) is the effective welding power, which is expressed as

$$Q=\eta \cdot P$$

(5)

where η and P represent welding efficiency and welding power, respectively.

The items on the right side of Equation (5) are derived as follows:

$$\begin{array}{l}{\displaystyle {\int }_{z_{i1}}^{z_t}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{r_1\left(z\right)}{q}_{m1}\left(r,z\right)}}}rdrd\theta dz\\ ={\displaystyle \frac{Q_0\pi \left(1-e^{-3}\right)}{3}}{\displaystyle \frac{z_t-z_{i1}}{\ln {\xi }_1}}\left({\xi }_1{r}_t^2-r_{i1}^2-{\displaystyle \frac{z_t-z_{i1}}{\ln {\xi }_1}}\left(\left({\displaystyle \frac{4z_t^3}{{p_1}^2}}+4{\displaystyle \frac{z_t}{p_1}}{k}_1\right){\xi }_1-\left({\displaystyle \frac{4z_{i1}^3}{{p_1}^2}}+4{\displaystyle \frac{z_{i1}}{p_1}}{k}_1\right)\right.\right.\\ \hspace{1em}-{\displaystyle \frac{z_t-z_{i1}}{\ln {\xi }_1}}\left(\left({\displaystyle \frac{12z_t^2}{{p_1}^2}}+4{\displaystyle \frac{k_1}{p_1}}\right){\xi }_1-\left({\displaystyle \frac{12z_{i1}^2}{{p_1}^2}}+4{\displaystyle \frac{k_1}{p_1}}\right)\right)\\ \left.\left.\left.\hspace{1em}-{\displaystyle \frac{z_t-z_{i1}}{\ln {\xi }_1}}{\displaystyle \frac{24}{{p_1}^2}}\left(\left(z_t{\xi }_1-z_{i1}\right)-{\displaystyle \frac{z_t-z_{i1}}{\ln {\xi }_1}}\left({\xi }_1-1\right)\right)\right)\right)\right)\end{array}$$

(6)

$$\begin{array}{l}{\displaystyle {\int }_{z_{i2}}^{z_{i1}}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{r_2\left(z\right)}{q}_{m2}\left(r,z\right)}}}rdrd\theta dz\\ ={\displaystyle \frac{Q_0\pi \left(1-e^{-3}\right)}{3}}\left(\left({\displaystyle \frac{z_{i1}^5}{5{p_2}^2}}+2{\displaystyle \frac{z_{i1}^3}{3p_2}}{k}_2+{k_2}^2{z}_{i1}\right)-\left({\displaystyle \frac{z_{i2}^5}{5{p_2}^2}}+2{\displaystyle \frac{z_{i2}^3}{3p_2}}{k}_2+{k_2}^2{z}_{i2}\right)\right)\end{array}$$

(7)

$$\begin{array}{l}{\displaystyle {\int }_{z_b}^{z_{i2}}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{r_3\left(z\right)}{q}_{m3}\left(r,z\right)}}}rdrd\theta dz\\ ={\displaystyle \frac{Q_0\pi \left(1-e^{-3}\right)}{3}}{\displaystyle \frac{z_b-z_{i2}}{\ln {\xi }_2}}\left(r_{i2}^2-{\xi }_2{r}_b^2-{\displaystyle \frac{z_b-z_{i2}}{\ln {\xi }_2}}\left(\left({\displaystyle \frac{4z_{i2}^3}{{p_1}^2}}+4{\displaystyle \frac{z_{i2}}{p_1}}{k}_1\right)-\left({\displaystyle \frac{4z_b^3}{{p_1}^2}}+4{\displaystyle \frac{z_b}{p_1}}{k}_1\right){\xi }_2\right.\right.\\ \hspace{1em}-{\displaystyle \frac{z_b-z_{i2}}{\ln {\xi }_2}}\left(\left({\displaystyle \frac{12z_{i2}^2}{{p_1}^2}}+4{\displaystyle \frac{k_1}{p_1}}\right)-\left({\displaystyle \frac{12z_b^2}{{p_1}^2}}+4{\displaystyle \frac{k_1}{p_1}}\right)\right.{\xi }_2\\ \left.\left.\left.\hspace{1em}-{\displaystyle \frac{z_b-z_{i2}}{\ln {\xi }_2}}{\displaystyle \frac{24}{{p_1}^2}}\left(\left(z_{i2}-z_b{\xi }_2\right)-{\displaystyle \frac{z_b-z_{i2}}{\ln {\xi }_2}}\left(1-{\xi }_2\right)\right)\right)\right)\right)\end{array}$$

(8)

where ξ_i represents the heat source coefficient.

After determining the required coefficients, the expression of the DPRC composite heat source model can finally be expressed as Equation (9):

$$q_m\left(r,z\right)=q_{m1}\left(r,z\right)+q_{m2}\left(r,z\right)+q_{m3}\left(r,z\right)$$

(9)

where $q_{m1}\left(r,z\right)$, $q_{m2}\left(r,z\right)$, and $q_{m3}\left(r,z\right)$ are as follows:

$$\left\{\begin{array}{c}{q}_{m1}\left(r,z\right)=Q_0\exp \left({\displaystyle \frac{\ln {\xi }_1}{z_t-z_{i1}}}\left(z-z_{i1}\right)\right)\exp \left(-3r^2/{\left({\displaystyle \frac{z^2}{p_1}}+k_1\right)}^2\right){\hspace{1em} z}_{i1}<z\le z_t\\ q_{m2}\left(r,z\right)=Q_0\exp \left(-3r^2/{\left({\displaystyle \frac{z^2}{p_2}}+k_2\right)}^2\right){\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z}_{i2}<z\le z_{i1}\\ q_{m3}\left(r,z\right)=Q_0\exp \left({\displaystyle \frac{\ln {\xi }_2}{z_b-z_{i2}}}\left(z-z_{i2}\right)\right)\exp \left(-3r^2/{\left({\displaystyle \frac{z^2}{p_3}}+k_3\right)}^2\right){\hspace{1em}\hspace{1em}z}_b⩽\le z\le z_{i2}\end{array}\right.$$

(10)

According to the above transient thermodynamic control equation, the equivalent heat exchange coefficient was added to the surface of the welded joint; then, the DPRC heat source model was applied. Before solving the temperature field, the thermal physical property parameters of the welded joint need to be obtained. The commonly used material performance software JmatPro V7.0 is used to obtain the thermophysical parameters of welded joints with different thicknesses, as shown in Figure 5.

3.2. Mechanical Equilibrium Equation

Based on the principle of virtual displacement, the system composed of the entire element (during the finite element calculation process, the welded joint is discretized into a large number of small elements) will reach a self-balancing state, and the balance relationship expression Equation (11) [26] established between the element force and displacement is

$${\left\{dF\right\}}^e+{\left\{dR\right\}}^e-{\left[K_d\right]}^e{\left\{d\delta \right\}}^e=0$$

(11)

where ${\left\{dF\right\}}^e$ represents the increment of the element node force, ${\left\{dR\right\}}^e$ represents the equivalent node increment of the element’s initial strain, ${[K_d]}^e$ represents the element stiffness matrix, and ${\left\{d\delta \right\}}^e$ represents the displacement increment of the element node.

The parameter matrix expressions in Equation (12) are

$${\left\{dR\right\}}^e={{\displaystyle \int \left[B\right]}}^T\left[D\right]\left[B\right]d\mathsf{\Omega }$$

(12)

$${\left[K_d\right]}^e={{\displaystyle \int \left[B\right]}}^T\left\{C\right\}dTd\mathsf{\Omega }$$

(13)

The above formula $[B]$ represents the matrix connecting the displacement and strain of the element node, that is ${\left\{d\epsilon \right\}}^e=\left[B\right]{\left\{d\delta \right\}}^e$ [27]; $[D]$ represents the stiffness matrix; and $\left\{C\right\}$ represents the temperature–stress matrix.

Therefore, based on the above element force balance matrix equation, the overall balance equation of the entire welded joint can be established, and its expression Equation (14) [28,29] is

$$\left\{dF\right\}+\left\{dR\right\}-\left[K_d\right]\left\{d\delta \right\}=0$$

(14)

When conducting thermo-elastoplastic finite element simulation on welded joints, without external forces, the entire welding process will reach a self-equilibrium state internally. At this point, the above equation can be simplified as Equation (15):

$$\left[K_d\right]\left\{d\delta \right\}=\left\{dR\right\}$$

(15)

3.3. Numerical Calculation Process

Thermo-elastoplastic finite element analysis of high-power laser penetration welded joints with different thicknesses was conducted in this paper. The thermal analysis and stress analysis in welding belong to transient thermal analysis and thermo-mechanical coupling analysis, respectively. Considering that the stress field has little effect on the temperature field, the unidirectional sequential coupling method in this paper is used to complete the calculation of the temperature and stress fields of the welded joint. The specific calculation process is shown in Figure 6, and the DPRC model is applied to the welded joint; the overall calculation process utilizes APDL (ANSYS Parametric Design Language) to achieve a fully parameterized solution.

Below is a detailed description of the calculation process requiring attention and the subsequent data processing procedures. Using Case 2 as an illustrative example, Figure 7 presents the three-dimensional dimensions of a 10 mm thick welded joint. In this figure, the welding direction is indicated by a purple arrow, while three critical points (P1 to P3) marked in red (with X-coordinates at 10 mm, 50 mm, and 90 mm, respectively) are strategically positioned to evaluate the stability of the welding temperature field. Additionally, points P2, P4, and P5 are designated for stress evolution analysis across Case 1 to Case 3. For comprehensive stress distribution analysis, data from lines L1, L2, and L3 are extracted from the upper, middle, and bottom surfaces of the welded joint, respectively, enabling a comparative examination of longitudinal and transverse stress variations through the thickness direction. To ensure computational stability and prevent rigid body displacement, boundary constraints are applied at three specified red points: the first point is fully constrained (Ux = Uy = Uz = 0), the second point is constrained in the Y and Z directions (Uy = Uz = 0), and the third point is constrained only in the Y direction (Uy = 0). It is important to note that these configuration settings remain consistent across all cases, with Case 1 and Case 3 maintaining identical parameters to those of Case 2.

As shown in Figure 8, the mesh of the weld and its vicinity has been refined to improve computational accuracy and save computational time. The total units in Case 1 to Case 3 are 18,000, 30,000, and 45,000, respectively. The total nodes are 21,917, 34,441, and 50,096, respectively; the minimum grid size is 0.25 mm × 0.5 mm × 1 mm. Solid70 and Solid185 units are used, respectively, in the thermo-mechanical analysis. Mechanical parameters obtained using the software JMatpro V7.0 calculation for subsequent finite element calculations are shown in Figure 9.

4. Results and Discussion

4.1. Thermal Analysis

(1) Temperature field validation analysis

Figure 10 illustrates a comparison between the geometry of the weld cross-section predicted by numerical calculation for the three cases and the experiment. It can be clearly observed that the simulation was in good agreement with the experimental results. At the same time, five key parameters for calculating the weld using the DPRC composite heat source model were extracted, including the upper radius (UR) r_t and height (UH) z_i1 of the upper parabolic rotating body, the middle cylindrical radius (MR) r_i, and the bottom radius (BR) r_b and height (BH) z_i2 of the lower parabolic rotating body. These parameters were compared with the experiment, as shown in Figure 10. According to relevant research, many studies have used simulation and experimental comparisons of weld cross-sections to evaluate the accuracy of temperature field calculations [25,30]. According to Figure 10, it can be observed that the r_t errors of Case 1 to Case 3 are 1.1%, –0.59%, and –1.23%, respectively. The maximum errors of z_i1, r_i, z_i2, and r_b were 2.13%, –2.63%, 0.299%, and –1.16%, respectively. The maximum error was not more than 5%. Moreover, it is clear that the maximum error of a 15 mm thick welded joint (Case 3) in

Table 2. The simulation and experimental key parameters for Case 3.

Parameters	Item	Case 3	DC Model [9]
rt (mm)	Experiment	1.63	1.63
	Simulation	1.61	1.88
	Errors Δ (%)	−1.23	15.3
zi1 (mm)	Experiment	4.22	-
	Simulation	4.31	-
	Errors Δ (%)	2.13	-
ri (mm)	Experiment	0.77	0.77
	Simulation	0.76	0.84
	Errors Δ (%)	−1.3	9.09
zi2 (mm)	Experiment	11.60	-
	Simulation	11.58	-
	Errors Δ (%)	−0.17	-
rb (mm)	Experiment	1.34	1.34
	Simulation	1.35	1.38
	Errors Δ (%)	0.75	2.98

calculated by the DC heat source model [9] is as high as 15.3%. Therefore, the DPRC composite heat source model proposed in this paper has higher calculation accuracy and can be further used in the thermo-mechanical analysis of thick plate welded joints.

(2) Effect of thickness on temperature–time-dependent curves

In order to study the temperature field stability of Case 1 to Case 3, the temperature history curves of points P1, P2, and P3 in three cases along the weld direction were extracted. Figure 11a–c show that the peak temperature of Case 1 to Case 3 gradually decreases with the increase in laser welding power and welding joint thickness. Moreover, the fluctuation amplitude of the peak temperature of points P1, P2, and P3 does not exceed 0.3 °C; therefore, based on the results of previous relevant studies [22,25], it can be inferred that their temperature field is stable.

(3) Influence of thickness on temperature gradient

The temperature history curves of points P2, P4, and P5 along the thickness direction in Case 1 to Case 3 were extracted, as shown in Figure 12a–c, and it can be found that the maximum temperature gradient is generated between points P2 and P5. It is clear that the maximum temperature gradient in Case 1 to Case 3 gradually decreases with the increase in laser welding power and welded joint thickness. This is consistent with the simulation and experimental results of the temperature field weld cross-section, indicating that the stability of the molten pool improves with the increase in laser welding power and welded joint thickness.

4.2. Longitudinal Residual Stress Analysis

As shown in Figure 13, the longitudinal RS distribution curves of Case 1 to Case 3 along the thickness direction paths L1, L2, and L3 were extracted for further quantitative analysis of the longitudinal stress distribution patterns of welded joints with different thicknesses along the thickness direction. As can be seen from Figure 13, the longitudinal stress distribution in Case 1 to Case 3 from the weld center to both sides along paths L1, L2, and L3 shows tensile and compressive stresses. The distribution trend first rises and then decreases near the weld and then slowly decreases from the HAZ to both sides. This means that when an equal volume heat is input, that is, as the laser welding power increases from 9 kW to 22.5 kW, the longitudinal RS distribution state and distribution trend of the welded joints along the thickness path L1, L2, and L3 are the same in the thickness range of 6 mm to 15 mm. However, it can be seen from Figure 13a,c that the longitudinal RS of the FZ along paths L1 and L3 in Case 1 to Case 3 decreases with the increase in laser welding power and welded joint thickness. The former decreases from 431.4 MPa to 333.05 MPa, with a decrease of 98.35 MPa, while the latter decreases from 419.8 MPa to 325.5 MPa, with a decrease of 94.3 MPa. According to Figure 13b, the longitudinal stress of the FZ along path L2 in Case 1 to Case 3 increases with the increase in laser welding power and welded joint thickness, and the peak longitudinal RS increases from 421.8 MPa to 464.8 MPa, with an increase of 43 MPa. It can be clearly observed that the changing trend in Figure 13b is completely opposite that in Figure 13a,c. With the increase in laser welding power and thickness, the longitudinal stress distribution in the weld gradually changes from a “U” shape to a “W” shape. This is primarily attributed to increased temperature gradients and faster cooling rates, leading to elevated levels of residual stresses [31]. It can be observed that the calculated values of longitudinal and transverse residual stresses based on the new heat source model in this paper are in good agreement with the simulation values. This indicates that the prediction of residual stresses in welded joints is accurate. Moreover, it can be seen from Figure 13a–c that the longitudinal stress reduction between HAZ and BM along paths L1, L2, and L3 in Case 1 to Case 3 is the largest, and the longitudinal compressive stress of path L2 in Figure 13a decreases by 98.35 MPa with the increase in thickness. In addition, the maximum longitudinal stress gradients between HAZ and BM in Figure 13a–c are 485.4 MPa, 482.83 MPa, and 505.7 MPa, respectively. Therefore, it can be inferred from Figure 13 that when an equal volume heat and the same welding speed are input, as the laser welding power and welded joints thickness continue to increase, the longitudinal stress peak value of the FZ along the path L2 in Case 1 to Case 3 will correspondingly increase; however, it is the opposite at paths L1 and L3.

4.3. Transverse Residual Stress Analysis

In order to quantitatively analyze the transverse stress distribution of welded joints with different thicknesses, as shown in Figure 14, the transverse RS distribution curves along the thickness direction paths L1, L2, and L3 in Case 1 to Case 3 were extracted. As shown in Figure 14, as the thickness of the welded joint increases, the trend in the distribution of transverse stress from the weld center to both sides along paths L1, L2, and L3 in Case 1 to Case 3 is basically the same. However, unlike the stress state in the WZ in Figure 14b, the stress state in Figure 14a,c gradually alters from tensile stress to compressive stress, and the distribution shape of the former gradually transforms from a “U” shape to a “W” shape, while the latter gradually changes from a “U” shape to an “M” shape. Consequently, it can be concluded that when an equal volume heat is input, the transverse stress distribution state and distribution trend of welded joints along the paths L1, L2, and L3 remain unchanged. At the same time, it can be seen from Figure 14a,c that the transverse RS of FZ along the upper surface path L1 and the lower surface path L3 from Case 1 to Case 3 increases with the increase in laser welding power and welded joint thickness. The former increases from 25.5 MPa to −138.1 MPa, with an increase of 163.6 MPa. The latter increased from 7.9 MPa to –136.6 MPa, with an increase of 163.6 MPa. As shown in Figure 14b, the transverse stress of FZ along the middle path L2 in Case 1 to Case 3 also increases with the increase in laser welding power and welded joint thickness. Its value increases from 101 MPa to 176.8 MPa, with an increase of 75.8 MPa. In addition, it can be seen from Figure 14a–c that the maximum decrease in the transverse stress between the HAZ and BM along paths L1, L2, and L3 in Case 1 to Case 3 occurs along path L2 in Figure 14a, where the transverse tensile stress decreases by 44.8 MPa. It can be inferred from Figure 14 that when the equal volume heat and the same welding speed are input, as the laser welding power and welded joint thickness continue to increase, subject to internally self-equilibrating residual stress within the system [25], the transverse stress peak value of FZ along paths L1, L2, and L3 in Case 1 to Case 3 will increase correspondingly.

4.4. Stress Evolution Analysis

To further study the effect of thickness changes in welded joints on node stress, as shown in Figure 15, the longitudinal and transverse stress–time-dependent curves of nodes P2, P4, and P5 along the thickness direction from Case 1 to Case 3 were extracted, respectively. For Case 1 to Case 3, due to the influence of the welded joint thickness, there is a significant time delay when the longitudinal and transverse stress of nodes P4 and P5 reaches its peak compared to node P2. According to the thermal analysis of welded joints with different thicknesses in Section 4.1, when equal volume heat and the same welding speed are input, the welded joint thickness has a remarkable impact on the temperature distribution along the thickness direction.

It can be clearly observed from Figure 15 that the longitudinal and transverse stress amplitude of nodes P2, P4, and P5 along the thickness direction in Case 1 to Case 3 is proportional to the peak temperature of each node in Figure 12a. As shown in Figure 15, the maximum and minimum changes in longitudinal stress amplitude are nodes P2 and P4 of Case 1, and their values are 4833.6 MPa and 1372.9 MPa, respectively. The maximum and minimum changes in transverse stress amplitude are still nodes P2 and P4 of Case 1, and their values are 4785.1 MPa and 1361 MPa, respectively. During the heating and early cooling stages of the welded joint, as shown in Figure 15, the longitudinal stress of nodes P2 in Case 1 to Case 3 and the transverse stress of nodes P2, P4, and P5 are basically the same, and they all undergo a process of first rising, then falling, and then rising again. The corresponding changes in the node stress state are tensile–compressive–tensile. At the same time, the longitudinal stresses of nodes P4 and P5 in Case 1 to Case 3 have undergone a process of rising–falling–rising–falling–rising, and their corresponding node stress states changed in the form of tension–compression–tension–compression–tension. It can be seen that when the same volume heat and the same welding speed are applied, an increase in the laser welding power and thickness does not alter the longitudinal and transverse stress at nodes P2, P4, and P5 along paths L1, L2, and L3 from Case 1 to Case 3. However, during the later cooling stage, due to the influence of thickness variation between Case 1 and Case 3 on the temperature gradient in the thickness direction, the longitudinal and transverse stresses at nodes P2, P4, and P5 along the thickness direction fluctuate to varying degrees. The transverse stress at node P2 along the thickness direction from Case 1 to Case 3 has the most significant influence, decreasing from 7.9 MPa in Case 1 to −136.6 MPa, with a decrease of −163.6 MPa.

5. Conclusions

In this paper, a new heat source model suitable for describing the weld morphology of high-power laser penetration welding joints in thick plates was proposed. Thermo-elastic plastic finite element analysis was carried out via thermo-mechanical coupling using full parametric programming. The effects of laser welding power and joint thickness on the temperature field and RS distribution were determined, respectively. The research results are summarized as follows:

(1) A DPRC model that could better describe the weld morphology of thick plate butt joints in single-pass full penetration laser welding was proposed. Based on the weld morphology of welded joints with different thicknesses obtained from metallographic tests, a double parabola rotation and cylinder combined heat source model (DPRC model) suitable for the weld morphology was proposed, and the theoretical derivation of the heat source model was completed.

(2) The calculation accuracy of the new heat source model was verified due to the good consistency between the weld section morphology calculated based on the DPRC heat source model and the experimental values. Simulation and experimental results indicate that the r_t errors of Case 1 to Case 3 are 1.1%, –0.59%, and –1.23%, respectively. It was determined that laser welding power and welded joint thickness significantly affect the peak temperature and maximum temperature gradient of the temperature field.

(3) The effects of laser welding power and plate thickness on the longitudinal and transverse re-stress distribution in the weld and its vicinity were investigated. It was observed that the longitudinal stress distribution trend in the weld gradually changed from a “U” shape to a “W” shape. The maximum longitudinal stress gradients between HAZ and BM in Figure 13a–c are 485.4 MPa, 482.83 MPa, and 505.7 MPa, respectively, while the transverse stress distribution trend gradually changed from a “U” shape to an “M" shape.

(4) The relationship between stress fluctuation amplitude and the temperature distribution in the weld zone was revealed. The stress state evolution law of nodes in the weld was analyzed, and it was found that due to the influence of thickness, the stress peak time of nodes P4 and P5 was significantly delayed compared with node P3. Moreover, the amplitude of changes in the longitudinal and transverse stress at nodes P2, P4, and P5 along the thickness direction from Case 1 to Case 3 is proportional to the peak temperature at each node.